
Notice that \(-2\bar{U}(i)^-\frac{Q(i)}{a}I_{\lbrace s\ge N\rbrace }\le 0\) ,
using similar method as Theorem 3.5 in [1]}, we can prove \(\bar{U}(t,i)^-=0\)  for a.e. \(t\in [0,M]\)  and all \(i\in \mathcal {M}\) .
This completes the proof.
\(\Box \)
